I am installing a new vanity in my bathroom but it's slightly larger than the previous one and hits the tile base board on the wall. I need to trim about 5 inches off this baseboard so I can put the vanity flush with the wall.

You could try using one of the oscillating tools or rotozip with a tile blade but you may find it easier to notch the back of the vanity to fit over the base tile. I don't know how high up the wall your tile goes but most vanities have room on the bottom to allow it without actually cutting into the cabinet.
You could also gently pry off the base and cut it to length.

1. Can you remove the bottom course of tile? If so, you could install the vanity then cut the new tile to fit perfectly. This will require some wall repair but the finished look will be precise. It means finding the exact same tile ... or opting for a contrasting tile as an accent strip.

2. Your only alternative would be to use a ceramic disk in an angle grinder to cut a perfect line along the tile and remove it. This requires a very strong, steady hand and patience. The tile could chip anyway. Minor chipping can be hidden with a bead of caulk or quarter round tile pieces along the border.

Good luck. I will be holding my breath! Here is a site that gives man helpful tips:

a 4" grinder + tile blade'll make the cut for you HOWEVER its VERY dusty,,, maybe the oscillating tool's the answer,,, they weren't invented when i had to cut ours,,, IF i had enough work for it, i'd buy 1

I chickened out and just cut it off the wall. I'm going to cut it to size and then mortar it back on. I just don't have a tile cutter. Any other tool can do the job? Is just one cut. Would a circular saw with a fine blade shatter it?